Measures of Center and Variation

Comparing Measures of

Center and Variation

Choosing the most appropriate measures

Appropriate Measures of Center and Variation

• The shape of the distribution determines which measure of center
  and variability are best.

• Use the mean and range for symmetrical distributions.

• Use the median and IQR for asymmetrical distributions or distributions that have outliers.

​Symmetrical= mirror image on both sides

​Skewed displays are asymmetrical, which means they don't look the same on both sides. They can be skewed right or skewed left. Look for the "tail". The stem plot has a tail on the "right".

Multiple Choice

What is the shape of the distribution shown?

• The distribution is skewed to the right.

• The distribution is skewed to the left.

• The distribution is approximately symmetrical.

• The distribution is bimodal.

Multiple Choice

What is the shape of the distribution?

• The distribution is approximately symmetrical.

• The distribution is skewed to the right.

• The distribution is skewed to the left.

• The distribution is approximately uniform.

Multiple Choice

What shape is the distribution?

• The distribution is approximately symmetrical.

• The distribution is skewed to the left.

• The distribution is skewed to the right.

• The distribution is bimodal.

Multiple Choice

What shape is the distribution?

• The distribution is skewed to the right.

• The distribution is skewed to the left.

• The distribution is approximately symmetrical.

• The distribution is approximately uniform.

Multiple Choice

What is the shape of this data?

• symmetric

• skewed right

• skewed left

• uniform

Multiple Choice

What is the shape of this bar graph?

• Skew right

• Skew left

• Symmetrical

• uniform

What is an outlier?

A data point that differs substantially from the other points.
The data point is far away from the other points.


Example:
1, 5, 8, 3, 2, 6, 8, 78 , 4

78 is an outlier because it is far away from the other points.

Definition:

Multiple Choice

Identify the outlier for the given data?
23, 34, 27, 7, 30, 26, 28, 31, 34

• 7
• 23
• 31
• 34

Multiple Choice

Identify the outlier for the given data?
23, 34, 27, 7, 30, 26, 28, 31, 34

• 7
• 23
• 31
• 34

Appropriate Measures of Center and Variation

• The shape of the distribution determines which measure of center
  and variability are best.

• Use the mean and range for symmetrical distributions.

• Use the median and IQR for asymmetrical distributions or distributions that have outliers.

​Symmetrical= mirror image on both sides

"Resistant" Measures: A statistic that is not greatly influenced or affected by outliers or skew

• Mean is not resistant to skew or outliers

• Use mean as the measure of center for symmetric distributions

• Median is resistant to skew or outliers

• Use mean as the measure of center for skewed distributions or outliers

I would the mean as a measure of center and the range for a measure of variation because the dot plot is symmetrical. Mean is not resistant.

​Example:

Multiple Choice

Which measure of center best represents this set of data?
73, 75, 71, 72, 76

• Mean because there is no outlier.

• Median because there is an outlier.

• Mode because there is a uniform distribution.

Multiple Choice

Which measure of center best represents this set of data?
12, 13, 11, 16, 31, 14, 10, 15

• Mean
• median
• mode

Drag and Drop

I would use ​ ​

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as the measure of center and ​

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for the measure of variation because the ​

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is ​

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.

Drag these tiles and drop them in the correct blank above

median

stem and leaf plot

skewed

IQR

mean

dot plot

symmetrical

Drag and Drop

I would use ​​ ​

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as the measure of center and ​ ​

Placeholder

for the measure of variation because the ​distribution is ​ ​

Placeholder

.

Drag these tiles and drop them in the correct blank above

median

skewed

IQR

mean

symmetrical

range

Drag and Drop

I would use ​​ ​ ​

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as the measure of center and ​ ​ ​

Placeholder

for the measure of variation because the distribution has an ​ ​

Placeholder

.

Drag these tiles and drop them in the correct blank above

median

IQR

mean

outlier

range

skew

symmetric

Open Ended

The stem and leaf plot shows the scores on a recent math test. The teacher wants to know how spread out the scores are.

Find the best measure (range or IQR) for the teacher to use and explain its meaning in the context of the situation.

Open Ended

The line plot shows the ages of applicants to orchestra camp. The camp director wants to know the average age of the camp applicants.

Find the best measure (mean or median) for the camp director to use and explain its meaning in the context of the situation.

Median and IQR

Median - the middle value in an ordered data set

●Roughly 50% of the data is below and above the median

●Measures the typical value for data in a skewed distribution

●NOT influenced by all data values

Interquartile Range (IQR) - the middle 50% of the data in a distribution

●Found by subtracting the 1st quartile (Q1) from the 3rd quartile (Q3)

●Measures the variability of a sample with a skewed distribution

Comparing Measures of Center

●

In a symmetric distribution, the mean and the median are
approximately the same.

●

In a right skewed distribution the mean tends to be greater than the
median.

●

In a left skewed distribution the mean tends to be less than the
median.

Drag and Drop

This distribution is ​

Placeholder

so we should use the ​

Placeholder

as a measure of center and the ​

Placeholder

as a measure of variability.

Drag these tiles and drop them in the correct blank above

right skewed

median

IQR

fairly symmetric

left skewed

mean

standard deviation

range

mode

Drag and Drop

This distribution is ​

Placeholder

so we should use the ​

Placeholder

as a measure of center and the ​

Placeholder

as a measure of variability.

Drag these tiles and drop them in the correct blank above

fairly symmetric

mean

range

left skewed

mode

median

IQR

right skewed

Multiple Choice

Which measure of center would we use with this data?

• Mean, since the data is fairly symmetric.

• Median, since the data is skewed.

• Standard deviation, since the data is symmetric.

• IQR, since the data is skewed.

Multiple Choice

Which measure of variation would we use with this data?

• Mean, since the data is fairly symmetric.

• Median, since the data is skewed.

• Standard deviation, since the data is symmetric.

• IQR, since the data is skewed.